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Random matrices have simple spectrum
Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple spectrum“. Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are distinct....
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254A, Supplement 2: A little bit of complex and Fourier analysis
We will shortly turn to the complex-analytic approach to multiplicative number theory, which relies on the basic properties of complex analytic functions. In this supplement to the main notes, we...
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254A, Notes 2: Complex-analytic multiplicative number theory
In Notes 1, we approached multiplicative number theory (the study of multiplicative functions and their relatives) via elementary methods, in which attention was primarily focused on obtaining...
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254A, Supplement 3: The Gamma function and the functional equation (optional)
In Notes 2, the Riemann zeta function (and more generally, the Dirichlet -functions ) were extended meromorphically into the region in and to the right of the critical strip. This is a sufficient...
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Long gaps between primes
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and I have just uploaded to the arXiv our paper “Long gaps between primes“. This is a followup work to our two previous papers (discussed in this...
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The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang...
In 1946, Ulam, in response to a theorem of Anning and Erdös, posed the following problem: Problem 1 (Erdös-Ulam problem) Let be a set such that the distance between any two points in is rational. Is it...
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